Evolution of Cyclic Mixmaster Universes with Non-comoving Radiation
EEvolution of Cyclic Mixmaster Universes with Non-comoving Radiation
Chandrima Ganguly and John D. Barrow
DAMTP, Centre for Mathematical Sciences,University of Cambridge,Wilberforce Rd., Cambridge CB3 0WA, U.K. (Dated: October 3, 2017)We study a model of a cyclic, spatially homogeneous, anisotropic, ‘mixmaster’ universe of Bianchitype IX, containing a radiation field with non-comoving (‘tilted’ with respect to the tetrad frame ofreference) velocities and vorticity. We employ a combination of numerical and approximate analyticmethods to investigate the consequences of the second law of thermodynamics on the evolution. Wemodel a smooth cycle-to-cycle evolution of the mixmaster universe, bouncing at a finite minimum,by the device of adding a comoving ‘ghost’ field with negative energy density. In the absenceof a cosmological constant, an increase in entropy, injected at the start of each cycle, causes anincrease in the volume maxima, increasing approach to flatness, falling velocities and vorticities,and growing anisotropy at the expansion maxima of successive cycles. We find that the velocitiesoscillate rapidly as they evolve and change logarithmically in time relative to the expansion volume.When the conservation of momentum and angular momentum constraints are imposed, the spatialcomponents of these velocities fall to smaller values when the entropy density increases, and viceversa. Isotropisation is found to occur when a positive cosmological constant is added because thesequence of oscillations ends and the dynamics expand forever, evolving towards a quasi de Sitterasymptote with constant velocity amplitudes. The case of a single cycle of evolution with a negativecosmological constant added is also studied.
I. INTRODUCTION
Cyclic models are popular alternatives to inflationary paradigms as candidates for a viable theory of theearly universe that avoids or mitigates the singularities in the simple Friedmann universes. For this to be asuitable theory, it must reproduce some of the successes of inflation. One of these is to produce a high degreeof isotropy at late times. In order to discuss the process of isotropisation, we will consider the most generalspatially homogeneous closed universe with a non-comoving velocity field. This generalises our earlier workwithout non-comoving velocities and will allow us to study the behaviour of velocities and dynamics of ageneral closed cyclic universe over many cycles.The simplest cyclic universes were constructed in dust-filled or radiation-filled closed Friedmann universes[1]. Using these simple models, Tolman was able to show that oscillating Friedmann universes with zerocosmological constant displayed successive cycles of increasing maximum size and duration, see also ref. [2]for a more general result. Later, this study was generalised to show that, when a positive cosmologicalconstant is included, Tolman’s cycles approach flatness but always come to an end: the dynamics ends ina state of expansion evolving towards a de Sitter universe [3]. If the entropy increase from cycle to cycle ismodest then his final state displays close proximity to flatness with a slight domination by dark energy (thecosmological constant stress) over cold dark matter (or radiation).These isotropic models are highly idealised, especially near the initial and final singularities, or expansionminima, of cyclic universes. We need to generalise them by studying the shape evolution of the most general,closed, cyclic, anisotropic universes. A start was made on this problem by considering anisotropic Kantowski-Sachs closed universes and Bianchi type I universes with a negative cosmological constant in ref. [3]. Thiswas generalised to the closed spatially homogeneous universe with comoving fluid velocities, of Bianchi typeIX, by the authors in ref. [4]. This ‘mixmaster’ cosmological model contains the closed Friedmann model as aspecial case but introduces several new factors, including anisotropic expansion rates (shear) and anisotropic3-curvature, which can change sign in the course of the evolution of the universe. This feature is intrinsicallygeneral relativistic and these models have no Newtonian counterparts. They allow us to study the evolutionof anisotropy over a sequence of cosmological cycles. In ref. [4] we showed that this evolution displayschaotic sensitivity to past conditions and successive maxima of a closed universe with increasing entropywill get larger and increasingly approach flatness, as in the isotropic case, but they will become increasinglyanisotropic at these growing maxima with respect to their expansion anisotropy (shear) and 3-curvatureanisotropy.In ref. [4], we studied the behaviour of a Bianchi IX universe with comoving matter velocities. Here, we a r X i v : . [ g r- q c ] O c t extend this by introducing the last remaining physical generalisation available to this metric – the inclusionof matter (or radiation) which moves with a 4-velocity that is not comoving with the tetrad frame. In thecontext of the orthonormal frame formalism, the system contains shear, anisotropic spatial curvature, andvorticity, [5]. The non-comoving fluid velocities are tilted with respect to normals to the hypersurfaces ofconstant density [6].There have been studies of non-comoving matter in other Bianchi types, such as Bianchi type VII , suchas in [7–10], but with a focus on the fate of primordial turbulence and the effects of collisionless particleson vortices. There has also been a study of the problem in a radiation-filled universe in ref. [11]. Here weextend these analyses, and those of cyclic universes, to include a Bianchi IX (‘mixmaster’) universe withnon-comoving radiation. We also include a comoving null energy condition (NEC) violating, or ‘ghost’, fieldwith p > ρ and ρ < t = 0 which will produce chaoticmixmaster oscillations [14] and thus avoids the inclusion of infinite mixmaster oscillations on approach tothe expansion minima. These oscillations are not likely to be physically relevant in classical cosmologicalevolution: if the universe bounces at the Planck time ( t pl ∼ − s ) then only a few mixmaster oscillationsare permitted up to the present ( t ∼ t pl ) because they occur in log-log of the comoving proper time.This makes our problem more tractable from a numerical perspective. Our aim is study the effect of non-comoving radiation in the case of a cyclic mixmaster universe with thermal entropy growth, in the presenceof both zero, positive and negative cosmological constant.In section 2, we set up the Einstein equations for the problem, with a brief background to the tetradformalism in Bianchi IX models, and give the energy-momentum tensor of the non-comoving radiation fieldthat we are introducing. We derive the equations of motion and the evolution equation for the velocitiesthat are normalised with the appropriate power of the energy density of radiation. In section 3, we discussthe qualitative effects of entropy increase on the cycle to cycle evolution of closed isotropic and anisotropicuniverses and identify a new effect of entropy increase introduced by the presence of non-comoving velocities.In section 4, we provide some analytic analysis of the type IX equations with velocities before presenting ourcomputational solutions of the Einstein equations with and without a cosmological constant of either signin section 5, and give our conclusions in section 6. II. THE SETUPA. The Einstein equations
For the purposes of studying the effect of non-comoving velocities in anisotropic closed cyclic universes,we choose the Bianchi IX universe. In general, when studying an n -dimensional spatially homogeneous,anisotropic cosmology, we consider a group of n linearly independent differential forms which remain invariantunder a group of simply transitive motions following ref. [15, 16], e aµ ( x (cid:48) ν ) dx (cid:48) µ = e aµ ( x ν ) dx µ , (1)where x (cid:48) µ and x ν are the coordinates in the transformed and the starting coordinate systems, respectively.We can then write down an invariant metric, ds = γ ab e aµ ( x (cid:48) ) e bν ( x (cid:48) ) dx (cid:48) µ dx (cid:48) ν = g µν ( x (cid:48) ) dx (cid:48) µ dx (cid:48) ν . (2)As the line element itself remains invariant under these transformations, we can also write ds = γ ab e aµ ( x ) e bν ( x ) dx µ dx ν = g µν ( x ) dx µ dx ν . (3)Considering the transformation between x (cid:48) µ and x µ , ∂x (cid:48) λ ∂x µ = e λa ( x (cid:48) ν ) e aµ ( x ρ ) , (4)and using the fact that double differentiation must commute under interchange of the indices µ and ν , ∂ x (cid:48) λ ∂x µ ∂x ν − ∂ x (cid:48) λ ∂x ν ∂x µ = 0 , (5)we can define the following commutation relation, e µd,α e αc − e µc,α e αd = C fcd e µf . (6)The C fcd are the structure constants of the Lie algebra, and obey the Jacobi identities: C fcd C gfe + C fde C gfc + C fec C gfd = 0 . (7)We can choose to work in a system of coordinates that is more suited to our purpose. We are dealingwith spatially homogeneous cosmologies and we are able to define coordinates on the spatial hypersurfacewhere t = const and the comoving proper time coordinate, t , will just measure the distance between parallelhypersurfaces. Thus we write the metric now as, ds = dt − g ik dx i dx k . (8)The simply transitive group of motions that leaves the differential forms invariant now acts on the three-spaces where t = const . We can then write out the Ricci tensor components in terms of the metric: R = (ln √− g )¨ + 14 g lm ˙ g mk g kj ˙ g jl , (9) R k = 12 g lm ( ˙ g lm ; k − ˙ g lk ; m ) , (10) R ij = R (cid:63)ij + 12 ¨ g ij −
12 ˙ g im g mk ˙ g kj + 12 ˙ g ij (ln √− g ) . (11)In our case we introduce the metric of the diagonal Bianchi IX universe, ds = dt − γ ab e aµ e bν dx µ dx ν , (12)where γ ab = diag[ a ( t ) , b ( t ) , c ( t ) ] . (13)Turning our attention now to the matter sector, we introduce the energy-momentum tensor for a perfectfield: T ab = ( ρ + p ) u a u b − pγ ab . (14)The 4-velocity of the perfect fluid with respect to our chosen tetrad frame is u a = ( u , u , u , u ) . (15)The relations of the components of the velocities with respect to the universe frame are given by, u a = e µa ¯ u µ , (16)where the ¯ u µ are the components of the 4-velocity of the fluid with respect to the universe frame. Weshall be working with the 4-velocity in the tetrad frame for consistency with the Ricci tensor, which is alsowritten in the tetrad frame for the purposes of this computation. The components of this 4-velocity obeythe normalisation, u − u − u − u = 1 . (17)Referring to [5], we find the following conditions on the energy-momentum tensor. The fluid vorticity ω ab iszero if and only if the spatial velocity components u i are zero. Thus, for the general case of non-comovingfields, we do indeed have vorticity in our system. Thus, with reference to the orthonormal frame formalism,we have non-zero shear σ ab , the curvature variables n ab as well as vorticity ω ab .For the purposes of our computation, we consider non-interacting perfect fluids, with ideal equation of state, p = ( γ − ρ, (18)and we can add their energy-momentum tensors together in the usual way. In our system, we includeradiation with γ ≡ γ r = 4 / γ ≡ γ g = 5 , avalue chosen simply for convenience in effecting a bounce. The densities and the pressures of the radiationand the ‘ghost’ field are given by ρ r , p r and ρ g and p g . The radiation field has velocities which are notcomoving in the tetrad frame of reference. We normalise the 4-velocity of the radiation field so that thenormalised velocity components are related to the velocity vector by v a = ( ρ r + p r ) /γ − / u a , and denotethe normalised velocity vector by v = ( v , v , v , v ) (19)In our case, for black body radiation, γ r = 4 / v a = ( ρ r + p r ) / u a . Considering energy-momentum conservation in the tetrad frame, we get the conservationof particle current, 1 √− g ∂∂x i ( √− gsu i ) = 0 , (20)where s is the entropy density. For radiation, s ∝ ρ / , this yields the conservation law, v a b c ( ρ r + p r ) = const ≡ L , (21)where we have labelled the constant L for consistency with reference [11].The second constraint equation for the components of velocity of the radiation field is v + v + v = Lδ (22)The constant L has the dimensions of length and the constant δ is dimensionless. Close to isotropy, when thespatial components of the velocity 4-vector are negligible, we have δ (cid:28)
1. For the case of small velocities ina near- Friedmann radiation-dominated universe, we see that their spatial components are constant. For thedust-dominated universe, the spatial components of the velocities fall as 1 /a where a ( t ) is the scale factorof the Friedmann universe and t is the comoving proper time.We have a further hydrodynamic equation of motion ( ∇ a T ab = 0), and 4-velocity normalisation [17] toemploy in what follows: ( p + ρ ) u k (cid:18) ∂u i ∂x k − u l ∂g kl ∂x i (cid:19) = − ∂p∂x i − u i u k ∂p∂x k (23) u i u i = 1 (24) B. Non-comoving velocities in a Bianchi type IX universe
We now ask what happens in an anisotropic, spatially homogeneous universes, with scale factors a, b, c ,when there are non-comoving velocities and vorticities. Suppose we first take the background expansion ofthe scale factors to have the same form as in the type IX universe without non-comoving velocities, that westudied in ref. [4]. In our earlier study without velocities we found a long period of evolution during theradiation era (before the curvature creates a slow-down of the expansion near the volume maximum) where,farfrom the expansion maximum, the scale factors evolve to a good approximation in a quasi-axisymmetricmanner, as a ( t ) = a t / [ln( t )] − / , b ( t ) = b t / [ln( t )] − / , c ( t ) = c t / ln( t ) . (25)Note that the volume, abc ∝ t / , evolves like the Friedmann model [11]. The logarithmic corrections arefamiliar in the study of anisotropic universes with anisotropic 3-curvatures, trace-free radiation stresses inthe presence of isotropic radiation, or long-wavelength gravitational waves [18]. They reflect the presenceof a zero eigenvalue when we perturb around the shear variables around the isotropic model whereas thevolume has a negative real eigenvalue.More generally, the effects of the velocities in the type IX radiation universe can be treated as test motionson an expanding radiation background governed by equations (20) and (21): a ( t ) = a t / ln λ ( t ) , b ( t ) = b t / ln µ ( t ) , c ( t ) = c t / ln ν ( t ) , (26a) λ + µ + ν = 0 , and λ, µ, ν constants, (26b) abc ∝ t / , (26c)where (26a) reduces to the particular case (25) when λ = µ = − / ν = 1.Ignoring spatial gradients with respect to time variations, and taking the diagonal scale factors to be a ( t ) , b ( t ) , and c ( t ), for a radiation-dominated universe ( p = ρ/ abcu ρ / = t / u ρ / = constant , (27) u α ρ / = constant ,α = 1 , , . (28)If we solve them as t → ∞ with λ < µ < ν, then the dominant component of u α is u = u /a , which gives u (cid:39) u u = ( u ) t − ln − λ ( t ), and we get the dominant late-time behaviours from (27)-(28): ρ ∝ ln λ ( t ) t , u u ∝ λ ( t ) , (29) T (cid:39) ρu u ∝ t ln λ ( t ) ∝ T (30) T (cid:39) ρu u ∝ ln λ − µ ( t ) t (31) T (cid:39) ρu u ∝ ln λ − ν ( t ) t (32)The corrections to the case with comoving velocities and zero vorticity are therefore only logarithmic intime during the radiation era. The scalar 3-velocity, has dominant asymptotic form V ≡ √ u α u α (cid:39) ln − λ ( t ) . (33)For the quasi-axisymmetric radiation-dominated phase of the type IX evolution, we take λ = µ = − / ν = 1 and we see that the stresses induced by the velocities grow logarithmically in time comparedto the other terms in the field equations (of order O (1 /t )) present when the velocities are comoving. Wesee that the diagonal stress-tensor components T ∝ T ∝ T ∝ t − ln / ( t ) fall off slower than t − as t → ∞ , while T ∝ t − ln − / ( t ) falls off faster than t − . We can see explicitly that the 3-velocity, V , isexpected to grow as ln / ( t ) in our approximation, which holds so long as the velocities are small enoughfor the perturbations not to disrupt the assumed (velocity-free) metric evolution (26a) and we are far fromthe expansion maximum. If there is an expansion maximum, then these asymptotic forms will be cut offwhen the approximate solution (25) breaks down and we need a numerical analysis to determine the detailedevolution in this regime, and from cycle to cycle. However, we expect the presence of non-comoving velocitiesto introduce changes to the analysis that was made for type IX cyclic universes in our work. [4].If we repeat this analysis in an isotropic de Sitter background with late-time scale factor evolution ap-proaching a = b = c = e H o t before the volume maximum, then the asymptotic behaviour of radiation is u α u α = const., u = const. , ρ r ∝ e − H t , and the new terms induced in the field equations by the non-comoving velocities do not grow at late times. However, we note that the velocities produce a constant tiltrelative to the normals to the surfaces of homogeneity and the asymptotic form at late times approachesde Sitter with a constant velocity field tilt (as also is seen in ref. [19]). In general, when the cosmologicalconstant, Λ ≡ H , is positive it will end the sequence of increasing oscillations in a cyclic closed universe,no matter how small its value, because the size of the universe will eventually become large enough for Λ todominate before a maximum is reached in some future cycle [3]. C. Equations of motion
In the type IX universe, the evolution equations for the velocities are as follows (where overdot is d/dt ):˙ v + v v v (cid:18) c − b (cid:19) (cid:18) L w − / a − b c ( a − b )( c − a ) (cid:19) = 0 , (34)˙ v + v v v (cid:18) a − c (cid:19) (cid:18) L w − / b − c a ( b − c )( a − b ) (cid:19) = 0 , (35)˙ v + v v v (cid:18) b − a (cid:19) (cid:18) L w − / c − a b ( c − a )( b − c ) (cid:19) = 0 , (36)where w ≡ ( ρ r + p r ). The evolution equations for the scale factors in cosmological time then become,(ln a )¨ + 3 H (ln a ) ˙ + 12 (cid:18) a b c − b c a − c a b (cid:19) + 1 a + 2 L (cid:18) a + c b ( c − a ) v − a + b c ( a − b ) v (cid:19) (37)= 2 (cid:18) w / v a + ρ r − p r − ρ g + p g (cid:19) (ln b )¨ + 3 H (ln b ) ˙ + 12 (cid:18) b c a − a b c − c a b (cid:19) + 1 b + 2 L (cid:18) b + a c ( b − a ) v − b + c a ( b − c ) v (cid:19) (38)= 2 (cid:18) w / v b + ρ r − p r − ρ g + p g (cid:19) (ln c )¨ + 3 H (ln c ) ˙ + 12 (cid:18) c a b − a b c − b a c (cid:19) + 1 c + 2 L (cid:18) c + b a ( b − c ) v − c + a b ( c − a ) v (cid:19) (39)= 2 (cid:18) w / v c + ρ r − p r − ρ g + p g (cid:19) These equations include a comoving ghost field ( ρ g ) and the non-comoving radiation field ( ρ r ). We give anapproximate analysis of the solutions to these equations during the radiation era, far from expansion minimaand maxima in the Appendix. We show there that the velocity components are constant up to logarithmicoscillatory factors during the era when the expansion dynamics are well approximated by 26a. III. INTRODUCING ENTROPY INCREASE
We want to investigate the effect of the non-comoving velocities on a closed cyclic type IX universe whenits radiation entropy increases from cycle to cycle, mirroring Tolman’s classic analysis [1]. The radiationentropy density is s ∝ ρ / . As in the earlier analysis made in ref. [4], we first consider a closed BianchiIX universe containing radiation, dust, and a ghost field but no cosmological constant. The ghost field hasnegative density and is dominant when the singularity is approached but dynamically irrelevant far fromthe initial and final singularities in each large cycle. It is included only to create a bounce at finite volume.This avoids evolution into the open interval of time around a curvature singularity at t = 0 during whichthe dynamics will be chaotic [14, 16, 20, 21]. For realistic choices of T ≈ − s as the start of classicalcosmology, there will be less than about 12 Mixmaster oscillations even if they continued all the way from T up to the present day [2, 22]. This is because the overall expansion scale changes rapidly with the numberof scale factor oscillations, which occur in log-log time.FIG. 1: Evolution of (a) the volume scale factors, and (b) the individual scale factors (left to right) withthe increase in entropy with time t in a Bianchi IX universe where the radiation is not comoving with thetetrad frame, as well as a comoving dust field, and a comoving ghost field to facilitate the bounce. Theblue starred, red dotted, and green lines correspond to the principal values of the 3-metric in the tetradframe, scale factors a ( t ), b ( t ) and c ( t ) respectively.
200 400 600 800 1000 1200 t V o l u m e Volume (a)
200 400 600 800 1000 1200 t I n d i v i d u a l s c a l e f a c t o r s a ( t ) b ( t ) c ( t ) (b) A. Effects of entropy increase
The effects of an increase in entropy from cycle to cycle of an isotropic oscillating closed universe werefirst considered by Tolman [1]. He showed that there would be an increase in expansion volume maxima andcycle length from cycle to cycle as a consequence of the second law of thermodynamics. The total energy ofthe universe is zero in each cycle and successive oscillations drive the universe closer and closer to flatness.If the dynamics are allowed to be anisotropic then we showed that, with Λ = 0, increasing entropyleads to the increase of volume maxima and cycle length in successive cycles but the anisotropy grows fromcycle to cycle in a manner that displays sensitive dependence on ‘initial’ conditions. We investigated thisdevelopment in the context of the Bianchi type IX universe with comoving fluid velocities– the most generalclosed spatially homogeneous universe containing an isotropic FLRW universe as a particular case [4]. Theaddition of Λ > ρ ∼ L ( abc ) − / , and the entropy density s ∝ ρ / for radiation. Increasing the entropy density fromcycle to cycle, means that L remains constant only per cycle but jumps to a higher value in the next cycle.Thus, the constraint equation (22) is valid in each cycle with the right hand side being equal to a new, largerconstant in subsequent cycles if there is entropy increase. A way of modelling this problem is to ensure thatthe constraint is imposed simultaneously with the injection of entropy at each minima. Thus, if we increasethe entropy, or in our case the energy density (as s ∝ ρ / ) by a factor ∆, then the normalised velocities v i = ρ / u i must be multiplied by a factor ∆ − / to keep the constraint equation (22) unchanged. Thus wesee that when the entropy increases, the velocities decrease as the evolution proceeds from cycle to cycle inaccord with the second law of thermodynamics.The sum of the square of the normalised velocities, ( ρ + p ) / u α , oscillates initially but eventually settlesdown to a nearly constant value with small oscillations around this value even as oscillations proceed tohigher and higher expansion maxima. In Figure 3, we show the constancy of this sum over one cycle. Wehave modelled the effects of radiation entropy, s , increase during a cycle of a closed universe by creating aFIG. 2: Evolution of the squares of velocities of non-comoving radiation with the increase in entropy withtime t in a Bianchi IX universe containing non-comoving radiation, as well as comoving dust and the ghostfields, the latter to facilitate the bounce. The velocity constraint (22) has been imposed. An increase inentropy(energy density) causes a decrease in the velocities and vice versa. Where necessary in the last twofigures, the figure has been magnified to capture the rapidly oscillating features of the plot. From left,clockwise, the entropy density ( s ∝ ρ / ),the square of the spatial components of the velocities, u , u and u are shown.
100 200 300 400 500 600 t . . . . . . . . E n t r o p y d e n s i t y o f r a d i a t i o n ρ ( t ) / (a)
100 200 300 400 500 600 t u ( t ) u ( t ) (b)
100 200 300 400 500 600 t u ( t ) u ( t ) t u ( t ) (c)
100 200 300 400 500 600 t u ( t ) u ( t ) t u ( t ) (d) sudden entropy increase at the start of each cycle . This produces the increase in the expansion maximumof successive cycles, first discovered by Tolman [1]We identify a new feature of isotropic, oscillating radiation universes: any non-comoving velocities andvorticities will diminish from cycle to cycle as the expansion maxima increase and flatness is approachedin accord with the second law of thermodynamics. For the anisotropic case, the overall trend in velocityevolution is oscillatory and is made more complicated. This is because we have shown that flatness isapproached with an increase in expansion maxima and the inclusion of non-comoving velocities changes We assume that the additional radiation entropy is at rest relative to the comoving frame so that we are not adding angularmomentum. The situation is analogous to the effect of quantum created particle at the Planck epoch on vortical motions,where the increase in inertia of created particles causes velocities to drop [23]
200 250 300 350 400 450 500 550 600 t T h e v e l o c i t y c o n s t r a i n t e q u a t i o n ρ ( t ) / ( u + u + u ) FIG. 3: The evolution of (22), the velocity constraint equation, over one cycleFIG. 4: Evolution of (a) the 3-curvature and (b) the shear with the increase of entropy with time t in aBianchi IX universe where the radiation is not comoving with the tetrad frame, also containing a comovingdust field and a comoving ghost field to facilitate the bounce.
200 400 600 800 1000 1200 t − . − . − . − . . . . . . - c u r v a t u r e -curvature (a)
200 400 600 800 1000 1200 t . . . . . . . . σ ( t ) σ ( t ) (b) the dependence of the energy density and hence of the entropy on the scale factors from the isotropic case(and the anisotropic case in the absence of these non-comoving velocities) [4]. Thus we can only observe aincrease/decrease in the velocities with a corresponding decrease/increase in the entropy. Aside from thiseffect, the evolutionary impact of the non-comoving velocities on the evolution in a cyclic radiation universefound in case with comoving velocities is only asymptotically logarithmic in time [4]. B. Evolution with non-comoving velocities
To study the behaviour of this model under the influence of non-comoving matter we assume that onlythe radiation field possesses non-comoving velocities (i.e. the ghost field is comoving). In the case of BianchiIX, we find that the scale factors do undergo a bouncing behaviour, see Figure 1a, as in the case without thenon-comoving velocities. The volume scale factor, abc , mimics the behaviour of cube of the scale factor inthe isotropic Friedmann case and shows an increase in height of its expansion maxima as the entropy of theconstituents is increased from cycle to cycle. The individual scale factors oscillate out of phase with eachother and with different expansion maxima, similar to their behaviour without the non-comoving velocities,see Figure 1b. However, the period of the volume oscillations is greater than in the comoving velocities case.Thus, the model takes longer to recollapse on average than in the comoving case, making each cycle lastlonger in comoving proper t time.The shear and the 3-curvature undergo oscillations which increase in amplitude and frequency near the0FIG. 5: (a), (b), and (c): Evolution of the squares of the 3-velocity components of non-comoving radiationwith the increase in entropy in time t in a Bianchi IX universe consisting of non-comoving radiation, aswell as comoving dust and the ghost fields, the latter to facilitate the bounce. Unlike in Figures 2b,2c and2d, the velocity constraint equation (22) has not been explicitly imposed. The evolution of u ( t ) and u ( t ) are highly oscillatory especially in the second cycle with very small time periods of oscillation, andto show this behaviour clearly, the plots are magnified and partly inset.
200 400 600 800 1000 1200 t u ( t ) u ( t ) (a) t u ( t ) u ( t ) t u ( t ) (b) t u ( t ) u ( t ) t u ( t ) (c) minima and do not appear to fall to smaller and smaller values, see Figures 4a and 4b.The velocity components themselves show oscillatory behaviour, see Figures 5a, 5b and 5c. However, theamplitude of their oscillations undergoes cyclic behaviour. The amplitudes of oscillations fall to their smallestvalues at the expansion minima of the scale factors. After the first oscillation, one of the velocity componentsstarts undergoing very small oscillations around a nearly constant value. We give an approximate analyticanalysis of this evolution in the Appendix.1 IV. THE EFFECTS OF A COSMOLOGICAL CONSTANT
FIG. 6: Evolution of (a) the shear, and (b) the 3-curvature (left to right) and the individual scale factorswith the increase in entropy with time t in a Bianchi IX universe where the radiation is not comoving withthe tetrad frame, containing a comoving dust field and a comoving ghost field to facilitate the bounce,together with a positive cosmological constant.
100 200 300 400 500 600 t . . . . . . σ ( t ) σ ( t ) (a)
100 200 300 400 500 600 t − . − . . . . . . - c u r v a t u r e -curvature (b) A. Positive cosmological constant ( Λ > ) Now we add a cosmological constant to the model. The effect of cosmological constant domination in thecase of comoving velocities was to cause the model to change from a cyclical behaviour to asymptoticallyde Sitter like expansion [4] (note that the cosmic no hair theorems [24, 25] do not cover the type IX casebecause the 3-curvature scalar can be positive).As in case with comoving velocities, the model is able to undergo cyclical behaviour until the maximagrow large enough for the cosmological constant to dominate at late times and then the dynamics approacha phase of quasi de Sitter expansion, see Figure 7a. The individual expansion rates oscillate while the modelis still undergoing cyclical behaviour but approach a constant value H = (cid:112) Λ / B. Negative cosmological constant ( Λ < ) Adding a negative cosmological constant results in the universe always recollapsing [26], as this is justanother null energy condition violating field. For the behaviour of the volume and individual scale factors,see Figures 10a and 10b.The ghost field allows the model to undergo more cycles of oscillation. As we are not introducing anincrease in entropy and all the cycles are of equal size, we shall focus on one cycle. The velocities all oscillateand increase with the volume of the universe. One of the velocities ( u ( t ) ) also oscillates but with smaller2FIG. 7: Evolution of (a) the volume scale factor, and (b) the individual directional Hubble rates (left toright) with the increase in entropy, and a positive cosmological constant, with time t in a Bianchi IXuniverse where the radiation is not comoving with the tetrad frame, also containing a comoving dust fieldand a comoving ghost field to facilitate the bounce. The blue starred, red dotted, and solid green linescorrespond to derviatives of the principal values of the 3-metric in the tetrad frame, Hubble rates ˙ a/a , ˙ b/b and ˙ c/c respectively. The model undergoes approach to de Sitter expansion when the cosmologicalconstant eventually dominates the dynamics after cycles become large enough to ensure this.
100 200 300 400 500 600 t V o l u m e V ( t ) (a)
100 200 300 400 500 600 t − . − . . . . H u bb l e r a t e s H a ( t ) H b ( t ) H c ( t ) (b) amplitude around a constant value. Only at the end of each cycle does this velocity component show anincrease in the amplitude of oscillations, see Figures 11a, 11b and 11c.The shear and the 3-curvature undergo oscillations, falling to their smallest values at the moments whenthe volume of the universe is at its highest, see Figures 9a and 9b. Again, we see the 3-curvature taking onnegative values when the dynamics are significantly anisotropic and positive values when close to isotropy. V. CONCLUSIONS
To complete the analysis of the shape of cyclic closed anisotropic universes, it is important to include theeffects of non-comoving matter. In the current analysis, we have extended the results of [4] by includinga radiation field that is not comoving with the reference tetrad frame. This tilted velocity field introducesvorticity, in addition to the shear and 3-curvature anisotropies, into the universe.We found that, as in the comoving case, the expansion maxima increases with increasing entropy of theconstituents from cycle to cycle, while the individual scale factors oscillate out of phase with each other.The overall dynamics approach flatness over many cycles but they become increasingly anisotropic. We finda new effect in oscillating universes with non-comoving velocities and vorticity. Over successive cycles ofentropy increase the conservation of momentum and angular momentum ensures that there is a decrease inthe magnitude of the velocities and vorticities in response to the increase of entropy. We modelled entropyincrease per cycle by adding entropy at the start of each cycle of a closed universe. We also included acomoving ghost field with negative energy density in order to create a bounce at finite expansion minimaand avoid the chaotic mixmaster regime as t → s ∝ ρ / ) produces a correspondingdecrease in the components of the non-comoving velocity, and vice versa.When we add a positive cosmological constant to a model containing radiation and a ghost field weconfirm that the oscillations are sustained until the cosmological constant dominates the dynamics, after3FIG. 8: Evolution of the squares of velocity components of non-comoving radiation with the increase inentropy with time t in a Bianchi IX universe consisting of non-comoving radiation, as well as a comovingdust field and a comoving ghost field to facilitate the bounce, and a positive cosmological constant. Thegraphs (a), (b) and (c) plot the squares of the spatial components of the 4-velocity in the tetrad frame, u ( t ) , u ( t ) , and u ( t ) , respectively. t u ( t ) u ( t ) (a) t u ( t ) u ( t ) (b) t u ( t ) u ( t ) (c) which the scale factors enter a period of quasi de Sitter expansion. The velocities oscillate with amplitudeincreasing with increasing scale factor as before, but after cosmological constant domination, the time periodof oscillations starts increasing, and they oscillate less rapidly, around a constant value. The asymptotic stateis de Sitter with a constant velocity field.When we add a negative cosmological constant we find there is always collapse, as expected. Studyingone cycle we see that the scale factors oscillate out of phase with each other. The velocities in two directionsoscillate with increasing amplitude as the volume increases but decrease again with decreasing volume. Thevelocity in the third direction, however, oscillates with very small amplitude around a constant value, onlyincreasing in oscillation amplitude at the end of each cycle when the volume is its smallest.We conclude that the inclusion of non-comoving velocities has the effect of increasing the time period ofthe oscillations of the model. The velocities oscillate rapidly per cycle but with increasing amplitude as thevolume of the universe increases, in at least two directions. In the third direction, the velocity oscillatesaround a constant value with very small amplitude, and hence remains nearly constant per cycle. It only4FIG. 9: Evolution of (a) the shear, and (b) the 3-curvature with the increase of entropy with time t in aBianchi IX universe where the radiation is not comoving with the tetrad frame, as well as comoving dustand ghost field, the latter to facilitate the bounce, in the presence of a negative cosmological constant.
50 100 150 200 250 300 t . . . . . . . . . . σ ( t ) σ ( t ) (a)
50 100 150 200 250 300 t − . − . . . . - c u r v a t u r e -curvature (b) FIG. 10: Evolution of (a) the volume scale factor and (b) the individual scale factors with t in the presenceof a negative cosmological constant in a Bianchi IX universe where the radiation is not comoving with thetetrad frame, and containing a comoving dust field and a comoving ghost field to facilitate the bounce. Theblue starred, red dotted and green solid lines correspond to the principal values of the 3-metric in thetetrad frame, scale factors a ( t ), b ( t ) and c ( t ), respectively.
50 100 150 200 250 300 t V o l u m e V ( t ) (a) t I n d i v i d u a l s c a l e f a c t o r s a ( t ) b ( t ) c ( t ) (b) increases in amplitude when the model collapses, before relapsing again to a nearly constant value duringthe next cycle. Our analysis has identified the principal ingredients of a general cyclic closed universe inthe case of spatial homogeneity. In a future work we will explore the effects of inhomogeneities on theseconclusions.5FIG. 11: Evolution of the squares of the velocities of non-comoving radiation with time t in a Bianchi IXuniverse consisting of non-comoving radiation, as well as comoving dust and ghost fields, the latter tofacilitate the bounce, and a negative cosmological constant. Plots (a), (b) and (c) show the squares of thespatial components of the 4-velocity in the tetrad frame, u ( t ) , u ( t ) , and u ( t ) , respectively. The highlyoscillatory behaviour of the velocity components with very short time period is captured by the magnifiedinsets in each of the plots.
50 100 150 200 250 300 t u ( t ) u ( t ) . . . . . t u ( t ) (a)
50 100 150 200 250 300 t u ( t ) u ( t ) . . . . . t u ( t ) (b)
50 100 150 200 250 300 t u ( t ) u ( t ) . . . . . t u ( t ) (c) Appendix A: Approximate analysis of the radiation era
We seek an approximate solution of the velocity evolution equations in the type IX model during theradiation era. In our earlier study [4] without velocities we found a long period of evolution during theradiation era (before the curvature creates slow-down of the expansion near the volume maximum) with thescale factors evolving to a good approximation in a quasi-axisymmetric manner during the radiation era, as a ( t ) = a t / [ln( t )] − / , b ( t ) = b t / [ln( t )] − / , c ( t ) = c t / ln( t ) . (A1)When the effects of the velocities in the Bianchi type IX radiation universe are small they can be treatedas test motions on an expanding radiation background governed by equations (20) and (21). We examine atypical case where we choose6 v = constant . This is consistent with the velocity evolution equation for v with 1 /a = 1 /b . In the approximation a (cid:29) b (cid:29) c and b > a c for large t from ((II C) and (II C)), the evolution equations for v and v reduce to:˙ v + v v v c (cid:18) − L w / (cid:19) = 0 , ˙ v − v v v c (cid:18) L b a w / (cid:19) = 0 . We assume non-relativistic velocities, so take v = 1, and note that w = ρ r + p r = 4 ρ r /
3. Since ρ r ∝ ( abc ) − / ∝ t − , we write w / = Mt , where M is a positive constant. Therefore the radiation entropy, s , depends on M via s ∝ ρ / r ∝ w / ∝ M / . Hence, we have approximately ˙ v + v v c t ln ( t ) (cid:18) − L tM (cid:19) = 0 , (A2)˙ v − v v c t ln ( t ) (cid:18) − L b tM a (cid:19) = 0 , (A3)where v is constant. At large times these equations are (and scaling a = b )˙ v = 2 L v v M c ln ( t ) ≡ Dv ln ( t ) , (A4)˙ v = − L v b v M c a ln ( t ) ≡ − Dv ln ( t ) , (A5)where D = 2 L v M c is a constant. Hence,we see immediately that v + v = E : E = constant . (A6)Since v = (cid:112) E − v , we have in (A4) ˙ v = − v ˙ v ( E − v ) − / = Dv ln ( t ) , (cid:90) dv (cid:112) E − v = − D (cid:90) dt ln ( t )Therefore, v = √ E sin (cid:18) − D (cid:90) dt ln ( t ) (cid:19) , and so, by (A6), we have v = √ E cos (cid:18) − D (cid:90) dt ln ( t ) (cid:19) . The components v and v therefore undergo bounded oscillations while v remains constant.We can get a better approx by keeping all the terms in (A2) and (A3). If we write them as˙ v + Av t ln ( t ) (1 − Bt ) = 0 , (A7)˙ v − Av t ln ( t ) (1 − Bt ) = 0 , (A8)then v + v = E , and hence we find a second order correction which confirms the oscillatory behaviourof he velocities with growing periods of oscillation: v = E / cos (cid:18) − t ) − B (cid:90) dt ln ( t ) (cid:19) v = E / sin (cid:18) − t ) − B (cid:90) dt ln ( t ) (cid:19) ACKNOWLEDGMENTS
J.D.B.is supported by the Science and Technology Facilities Council (STFC) of the United Kingdom. C.G.is supported by the Jawaharlal Nehru Memorial Trust Cambridge International Scholarship. C.G. wouldalso like to thank Bogdan V. Ganchev for useful discussions. [1] Richard C Tolman. On the theoretical requirements for a periodic behaviour of the universe.
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